The Green’s functions for peridynamic non-local diffusion

Abstract

In this work, we develop the Green's function method for the solution of the peridynamic non-local diffusion model in which the spatial gradient of the generalized potential in the classical theory is replaced by an integral of a generalized response function in a horizon. We first show that the general solutions of the peridynamic non-local diffusion model can be expressed as functionals of the corresponding Green's functions for point sources, along with volume constraints for non-local diffusion. Then, we obtain the Green's functions by the Fourier transform method for unsteady and steady diffusions in infinite domains. We also demonstrate that the peridynamic non-local solutions converge to the classical differential solutions when the non-local length approaches zero. Finally, the peridynamic analytical solutions are applied to an infinite plate heated by a Gauss source, and the predicted variations of temperature are compared with the classical local solutions. The peridynamic non-local diffusion model predicts a lower rate of variation of the field quantities than that of the classical theory, which is consistent with experimental observations. The developed method is applicable to general diffusion-type problems.

Generalized integral equation method for an elliptic nonlocal equation in measure space

“…Research on the Green's function for nonlocal models such as peridynamics and elasticity can be found in [27,30,31,33]. The discrete Green's function with asymptotically compatible quadrature-based finite difference scheme was presented in [7], which can be taken as a nonlocal analog for local PDEs [4].…”

Analysis of heat transfer and water flow with phase change in saturated porous media by bond-based peridynamics

“…For the known pressure field, that is defined by the solution of a system of linear equations (44), we can define the vector of water flux u(x, t) by the numerical solution of the relation (26). It is suitable to find the components of this vector separately, for that, the following system can be solved:…”

“…Bond-based formulations of transient heat conduction were originally developed in [23,16], see also [24,25]. Further developments of the bond-based PD heat transfer models were discussed in [26,27,28]. However, to date, the phase change phenomenon has not been included in the developments by this approach.…”

A Review of Peridynamics (PD) Theory of Diffusion Based Problems

“…Later Jabakhanji and Mohtar [11] extended the work of [10] to transient moisture flow in unsaturated, anisotropic, and heterogeneous soils in PD framework. A PD diffusion model by employing Green's function has been developed by [12]. In this study, the steady-state and transient PD Green's functions were derived and implemented to 2D infinite plate which is heated by a Gauss source.…”